_{1}

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The Einstein-Podolsky-Rosen paradox is resolved dynamically by using spin-dependent quantum trajectories inferred from Dirac’s equation for a relativistic electron. The theory provides a practical computational methodology for studying entanglement versus disentanglement for realistic Hamiltonians.

The Einstein-Podolsky-Rosen paradox [

The Pauli exclusion principle, which is fundamental for fermion structure and collision problems, states that each fermion in an ensemble must have a unique set of four quantum numbers three for space and one for spin. For example a pair of electrons can occupy the same spatial orbital only if they have opposite spin states. The canonical examples are the singlet and triplet states (for upper and lower signs respectively) of the helium atom or of the hydrogen molecule,

where the arguments refer to the 3-space position vectors of electrons 1 and 2. The second term in square brackets comprises up (alpha) and down (beta) spin states such that the 2-electron spin state is 0 (singlet state) or 1 (triplet state) for upper and lower signs respectively. This point is obvious if the orbitals labeled a and b are identical such that fermions 1 and 2 occupy the same spatial orbital for the singlet state (upper sign) while the first term in square bracket vanishes for the triplet state (lower sign) since the Pauli exclusion principle is violated for this case.

The singlet state is a canonical example of an entangled state since the two electrons cannot be separated spatially and appear therefore to transfer information between themselves instantaneously. In a way the mysterious nature of the entangled state is illusory due to the incompleteness of the physical theory itself. Firstly the theory is for stationary states with no dynamical information whatsoever between the two correlated electrons. Second Schroedinger theory is spinless such that the 2-electron state written in Equation (1) is an ad hoc construction based on experimental observation. It is true that the symmetry of the Schroedinger Hamiltonian with respect to the permutation of electron coordinates allows for pairs of spatial states in the first square bracket to have either even (upper sign) or odd permutation symmetry on the exchange of electrons. Nevertheless the spin dependence of the 2-electron wave function has a totally phenomenological origin such that no a priori physical theory exists to explain why the state with total spin angular momentum of 0 is an entangled state while the state with total spin angular momentum of 1 is an unentangled state.

Ironically the paradox is resolved by Einstein’s own theory of special relativity. Dirac discovered the correct quantum theory of special relativity for a fermion, which makes it possible to explain both the explicit dynamical nature of entanglement and its dependence on the spin state of a fermion.

The form of Dirac theory which makes this detailed understanding possible is outlined below. First I postulate that a correct dynamical theory for a relativistic electron interacting with other relativistic electrons can be had by replacing the classical relativistic equation of motion for each electron by Dirac’s equation as follows [

where

Quantum eigentrajectories in the z direction. Inner curves: R = 1.4 au (R is the internuclear distance). Outer curves: R = 3.0 au. The inner curves show how two electrons with opposite spin states correlate and entangle with increasing time and eventually find the region of covalent bonding located between the two protons fixed at au, while the outer curves show that the electrons remain in the vicinity of the separated atoms for all times. The eigentrajectories are calculated from Equation (5) in the nonrelativistic limit using eigenfunctions for the up and down spin states for the state of H_{2}

electrons equations of motion analogous to Equation (2) would be written for the primed-variable electron whose interaction with the other electron would now be expressed using the unprimed variable. Notice the pas-

sage from classical to quantum dynamics of Coulomb’s Law

any two electrons whose trajectories are at

The quantum trajectory is calculated for the unprimed-variable electron as follows. First this electron’s velocity field

where

and finally by finding the quantum expectation value of the position field,

and similarly for the primed electron.

In the nonrelativistic regime of electron velocity the current is evaluated in the nonrelativistic limit using

Quantum eigentrajectories showing how the two electrons of H_{2} correlate but remain unentangled with increasing time in the formation of an antibonding state. Solid: spin-up electron. Dotted: spin- up electron. The eigentrajectories are calculated from Equation (5) in the nonrelativistic limit using eigenfunctions for two parallel spin states of the state of H_{2}

spectively. Written out explicitly in terms of the large component the current given by Eq. (3) becomes

where we have used

from which the identities useful in evaluating the current can be inferred,

Written out explicitly for up (upper sign) or down (lower sign) spin states

The current in the nonrelativistic regime is

The first term on the right side of Equation (9), which is independent of spin, is contributed by Schroedinger theory, while the second and third terms are contributed uniquely by Dirac theory. Notice that the current and therefore a quantum trajectory scale like all of the other Schroedinger contributions, namely as c^{0} and not as c^{-2}, which have been dropped in the Schroedinger limit of Dirac’s equation. It is found in [

Notice finally that the first-principles understanding of Fermi-Dirac statistics makes available to us a new highly practical computational methodology in which one needs an efficient, accurate solver for the 3D time- dependent Schroedinger equation and an efficient, accurate, energy-conserving integrator for the quantum trajectories. Configuration interaction (CI) calculations are obviated since the time-dependent solution is a superposition of ground and excited states. One should not fuss that the electron-electron Coulomb potential has a mixed evaluation using an independent position variable for one electron and a dependent position variable for the other electron: quantum mechanics allows us latitude to calculate the inverse distance between two point particles as long as it is calculated wave mechanically and not deterministically. The mathematical bête noir of conventional time-independent many-electron quantum theory is of course the electron-electron Coulomb potential calculated as an inverse distance using independent position vectors for both electrons. Quantum mechanics does not require us to seek a single wave function for N electrons instead of N wave functions for N electrons, and the former appears to be an accident of the additivity of the Schroedinger Hamiltonian leading to a vast literature on independent-electron approximation methods and on scholastic research on density functionals in which angels are replaced by orbitals. Except for the Bethe-Salpeter equation for two fermions, relativistic invariance is satisfied by a one-body Dirac equation in 4-space: three spatial variables and the scaled time ct. Hence in Dirac theory it is natural to write N wave functions for N fermions as in Equation (2) instead of one wave function for N fermions. As long as the electron-electron potential is written as an exact instantaneous interaction in 3-space and the time, then both electron exchange-correlation and its corollary Fermi-Dirac statistics will be dynamically achieved.

In this paper, I have demonstrated quantum entanglement and disentanglement (

The author is grateful to T. Scott Carman for supporting this work. He is grateful to Professor John Knoblock of the University of Miami for the seminal discussion. This work was performed under the auspices of the Lawrence Livermore National Security, LLC, (LLNS) under Contract No. DE-AC52-07NA27344.